What are some of the changes in heart rate and breathing rate in the human body during excercise and the rest period right after excercise? What responses are made by the body to try to maintain homeostasis?

Considering that we are physically and physiologically healthy, when we move more than the normal, we breathe faster and our heart rate increases too. As our body returns to its usual condition, RR and HR are also normalized.

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During exercise, your body requires an increased amount of oxygen (your cells are respiring at an increasing rate to produce ATP, of which oxygen is essential in the creation of). Therefore, to compensate for this lack of oxygen, you breath faster and your heart pumps faster (to get the oxygen to the cells). After you exercise, your body will eventually reach homeostasis.

Also, the production of ATP produces heat. In order to maintain homeostasis, your body has mechanisms to get rid of the heat, sweating, vasodilation, etc.

Yes. In addition, soon after vigorous exercises, your breathing rate and heart beat rate are still higher than normal. It is to repay the oxygendebt due to anaerobic respiration. During vigorous exercises, cells in muscles carry out anaerobic respiration to get extra energy. However, lactic acid is released as waste product and then transferred to blood. Extra oxygen is required to oxidize the lactic acid so as to normalize the pH level in blood.